Simplify The Expression: $(5 \sqrt{3} - \sqrt{2})(2 \sqrt{3} + \sqrt{2}$\]

Introduction

In this article, we will simplify the given expression: $(5 \sqrt{3} - \sqrt{2})(2 \sqrt{3} + \sqrt{2})$. This involves using the distributive property and the properties of radicals to simplify the expression.

Understanding the Expression

The given expression is a product of two binomials, each containing a radical. The first binomial is $5 \sqrt{3} - \sqrt{2}$, and the second binomial is $2 \sqrt{3} + \sqrt{2}$. To simplify this expression, we will use the distributive property, which states that for any real numbers $a$, $b$, and $c$, $a(b + c) = ab + ac$.

Step 1: Apply the Distributive Property

To simplify the expression, we will apply the distributive property to each term in the first binomial. This means that we will multiply each term in the first binomial by each term in the second binomial.

$(5 \sqrt{3} - \sqrt{2})(2 \sqrt{3} + \sqrt{2})$

Using the distributive property, we can rewrite this expression as:

$5 \sqrt{3}(2 \sqrt{3} + \sqrt{2}) - \sqrt{2}(2 \sqrt{3} + \sqrt{2})$

Step 2: Simplify Each Term

Now that we have applied the distributive property, we can simplify each term in the expression.

$5 \sqrt{3}(2 \sqrt{3} + \sqrt{2}) - \sqrt{2}(2 \sqrt{3} + \sqrt{2})$

Using the distributive property again, we can rewrite this expression as:

$10 \sqrt{3} \sqrt{3} + 5 \sqrt{3} \sqrt{2} - 2 \sqrt{3} \sqrt{2} - \sqrt{2} \sqrt{2}$

Step 3: Simplify Radicals

Now that we have simplified each term, we can simplify the radicals.

$10 \sqrt{3} \sqrt{3} + 5 \sqrt{3} \sqrt{2} - 2 \sqrt{3} \sqrt{2} - \sqrt{2} \sqrt{2}$

Using the property of radicals that $\sqrt{a} \sqrt{b} = \sqrt{ab}$, we can rewrite this expression as:

$10 \sqrt{9} + 5 \sqrt{6} - 2 \sqrt{6} - 2$

Step 4: Simplify the Expression

Now that we have simplified the radicals, we can simplify the expression.

$10 \sqrt{9} + 5 \sqrt{6} - 2 \sqrt{6} - 2$

Using the property of radicals that $\sqrt{a^2} = a$, we can rewrite this expression as:

$10(3) + 5 \sqrt{6} - 2 \sqrt{6} - 2$

Step 5: Simplify the Final Expression

Now that we have simplified the expression, we can simplify the final expression.

$10(3) + 5 \sqrt{6} - 2 \sqrt{6} - 2$

Using the distributive property, we can rewrite this expression as:

$30 + 3 \sqrt{6} - 2$

Conclusion

In this article, we simplified the given expression: $(5 \sqrt{3} - \sqrt{2})(2 \sqrt{3} + \sqrt{2})$. We used the distributive property and the properties of radicals to simplify the expression. The final simplified expression is $30 + 3 \sqrt{6} - 2$.

Final Answer

The final answer is $\boxed{28 + 3 \sqrt{6}}$.

Key Takeaways

• The distributive property can be used to simplify expressions involving radicals.
• The properties of radicals can be used to simplify expressions involving radicals.
• Simplifying expressions involving radicals can be a complex process, but it can be broken down into smaller steps.

Common Mistakes

• Failing to apply the distributive property when simplifying expressions involving radicals.
• Failing to simplify radicals when simplifying expressions involving radicals.
• Failing to check the final expression for simplification.

Real-World Applications

Simplifying expressions involving radicals has many real-world applications, including:

• Physics: Simplifying expressions involving radicals is essential in physics, particularly in the study of wave motion and electromagnetic theory.
• Engineering: Simplifying expressions involving radicals is essential in engineering, particularly in the study of electrical circuits and mechanical systems.
• Computer Science: Simplifying expressions involving radicals is essential in computer science, particularly in the study of algorithms and data structures.

Conclusion

Introduction

In our previous article, we simplified the expression: $(5 \sqrt{3} - \sqrt{2})(2 \sqrt{3} + \sqrt{2})$. In this article, we will answer some frequently asked questions about simplifying expressions involving radicals.

Q: What is the distributive property?

A: The distributive property is a mathematical property that states that for any real numbers $a$, $b$, and $c$, $a(b + c) = ab + ac$. This property can be used to simplify expressions involving radicals.

Q: How do I apply the distributive property to simplify expressions involving radicals?

A: To apply the distributive property to simplify expressions involving radicals, you need to multiply each term in the first binomial by each term in the second binomial. This means that you will have to multiply each term in the first binomial by each term in the second binomial.

Q: What are some common mistakes to avoid when simplifying expressions involving radicals?

A: Some common mistakes to avoid when simplifying expressions involving radicals include:

• Failing to apply the distributive property when simplifying expressions involving radicals.
• Failing to simplify radicals when simplifying expressions involving radicals.
• Failing to check the final expression for simplification.

Q: How do I simplify radicals when simplifying expressions involving radicals?

A: To simplify radicals when simplifying expressions involving radicals, you need to use the properties of radicals. The properties of radicals state that $\sqrt{a} \sqrt{b} = \sqrt{ab}$ and $\sqrt{a^2} = a$. You can use these properties to simplify radicals and arrive at a final simplified expression.

Q: What are some real-world applications of simplifying expressions involving radicals?

A: Simplifying expressions involving radicals has many real-world applications, including:

• Physics: Simplifying expressions involving radicals is essential in physics, particularly in the study of wave motion and electromagnetic theory.
• Engineering: Simplifying expressions involving radicals is essential in engineering, particularly in the study of electrical circuits and mechanical systems.
• Computer Science: Simplifying expressions involving radicals is essential in computer science, particularly in the study of algorithms and data structures.

Q: How do I check the final expression for simplification?

A: To check the final expression for simplification, you need to make sure that you have applied the distributive property and simplified radicals correctly. You can do this by checking the final expression for any errors or simplifications that may have been missed.

Q: What are some tips for simplifying expressions involving radicals?

A: Some tips for simplifying expressions involving radicals include:

• Make sure to apply the distributive property correctly.
• Make sure to simplify radicals correctly.
• Check the final expression for simplification.
• Use the properties of radicals to simplify radicals.
• Break down the process into smaller steps.

Conclusion

In conclusion, simplifying expressions involving radicals is a complex process that requires a deep understanding of the properties of radicals and the distributive property. By answering some frequently asked questions about simplifying expressions involving radicals, we can gain a better understanding of the process and arrive at a final simplified expression.

Final Answer

The final answer is $\boxed{28 + 3 \sqrt{6}}$.

Key Takeaways

• The distributive property can be used to simplify expressions involving radicals.
• The properties of radicals can be used to simplify expressions involving radicals.
• Simplifying expressions involving radicals can be a complex process, but it can be broken down into smaller steps.
• Checking the final expression for simplification is essential to ensure that the expression has been simplified correctly.

Common Mistakes

• Failing to apply the distributive property when simplifying expressions involving radicals.
• Failing to simplify radicals when simplifying expressions involving radicals.
• Failing to check the final expression for simplification.

Real-World Applications

Simplifying expressions involving radicals has many real-world applications, including:

• Physics: Simplifying expressions involving radicals is essential in physics, particularly in the study of wave motion and electromagnetic theory.
• Engineering: Simplifying expressions involving radicals is essential in engineering, particularly in the study of electrical circuits and mechanical systems.
• Computer Science: Simplifying expressions involving radicals is essential in computer science, particularly in the study of algorithms and data structures.